Development of a method for the

measurement of long-range 13C–1H coupling

constants from HMBC spectra

Richard A E Edden and James Keeler ∗University of Cambridge, Department of Chemistry, Lensfield Rd, Cambridge,

CB2 1EW, U.K.

Abstract

This paper describes a number of improvements to a method, developed in thislaboratory and described in J. Magn. Reson. 85 (1989) 111-113, which makes itpossible to determine values of long-range 13C–1H coupling constants from HMBCspectra. Firstly, it is shown how pulsed-field gradients can be introduced into theHMBC experiment without perturbing the form of the cross-peak multiplets; a one-dimensional version of the experiment is also described which permits the rapidmeasurement of a small number of couplings. Secondly, the experiment is modifiedso that one-bond and long-range cross peaks can be separated, and so that theone-bond cross peaks have more reliable intensities. Finally, it is shown how theseone-bond cross peaks can be used to advantage in the fitting procedure.

Key words: HMBC, long-range C–H coupling, fitting, MBOBPACS:

1 Introduction

The HMBC (heteronuclear multiple bond correlation) [1] experiment has be-come the standard way of detecting the presence of long-range 13C–1H cou-plings in small- to medium-sized molecules. However, from such a spectrumit is not at all straightforward to determine the numerical value of the cou-pling constant. The difficulty derives from the fact that the multiplets in theproton (F2) dimension have complex phase properties due to the evolution of

∗ Corresponding authorEmail address: James.Keeler@ch.cam.ac.uk (James Keeler).URL: www-keeler.ch.cam.ac.uk (James Keeler).

Preprint submitted to Elsevier Science 6 August 2003

proton–proton couplings and proton chemical shifts (offsets); in addition, thelong-range coupling appears as an anti-phase splitting. It is thus not possibleto identify, from the multiplet, a splitting which corresponds to the value ofthe long-range coupling constant.

Previous work from this laboratory [2] introduced a fitting procedure whichmakes it possible to determine a value for the long-range coupling constantfrom an HMBC multiplet. The procedure works by comparing the HMBC mul-tiplet with the corresponding multiplet from a conventional proton spectrum(called the template); a least-squares fitting procedure, involving just two pa-rameters, yields a value for the long-range C–H coupling. If the conventionalproton spectrum is crowded, the proton multiplets needed for the fitting pro-cedure can be obtained from two-dimensional spectra provided that the mul-tiplets are in-phase and otherwise undistorted. For example, such multipletscan be obtained from TOCSY spectra [3] in which the anti-phase dispersivecontributions have been suppressed effectively [4].

In this paper we describe a number of developments of this basic approach.Firstly, the HMBC experiment is updated by the use of gradient pulses so thathigh-quality data can be obtained; we also discuss the problem of phasing suchspectra. Secondly, an experimental approach is described which makes it pos-sible to obtain, from one set of experimental data, both the HMBC spectrumand a complete one-bond correlation spectrum from which templates can beobtained. Obtaining the templates in this way is convenient as, compared tothe conventional proton spectrum, the likelihood of overlap is reduced. Wealso introduce a new way in which these one-bond cross peaks can be usedin the fitting procedure. The utility of the whole approach is illustrated bymeasuring the values of numerous long-range couplings in strychnine.

Finally, we introduce a one-dimensional version of the HMBC experimentwhich results in multiplets that can be analysed in the same way as those fromthe two-dimensional experiments. In cases where only a few couplings are ofinterest, a selective experiment such as this may be an attractive alternativeto recording a two-dimensional experiment.

The fitting procedure itself is not computationally demanding and can beimplemented on a desk-top computer. A program, designed to work withinBruker’s XWIN-NMR c© and available without charge from the authors’ website, has been developed to carry out the fitting procedure.

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2 The basic HMBC experiment and fitting procedure

The way in which the fitting procedure works and the required modificationsto the HMBC experiment have been described in the earlier work from thislaboratory [2]. However, it is useful to repeat these here as they form the basisfor the description of subsequent modifications, which are the topic of thispaper.

[Fig. 1 about here.]

The pulse sequence for the basic HMBC experiment is shown in Fig. 1 (a).Proton magnetization excited by the first pulse evolves during the preparationdelay, ∆, to become anti-phase with respect to the long-range C–H coupling;this anti-phase magnetization is converted into heteronuclear multiple quan-tum coherence by the first 13C pulse. The coherence evolves for time t1 duringwhich it acquires a phase label according to the 13C offset; the evolution ofthe proton offset during t1 is refocused by the centrally-placed 180 pulse onproton. Finally, the second 13C pulse returns the multiple quantum coher-ence to proton magnetization which is anti-phase with respect to the (active)long-range C–H coupling.

Although the proton 180 pulse refocuses the evolution of the proton offset overthe time t1, both the offset and any proton–proton couplings evolve throughoutthe delay ∆, thus giving rise to a phase modulation of the observed signal. As∆ has to be comparable with 1/(2 nJCH), the evolution of any proton-protoncouplings present is certainly significant. It is this modulation, combined withthe presence of the anti-phase splitting with respect to the C–H coupling, thatgives the HMBC multiplet its complex phase properties.

For the fitting procedure to work correctly, it is essential that the phase mod-ulation in the HMBC experiment is identical to that which would arise froma simple delay ∆. In the basic HMBC experiment this is not the case, due tothe the presence of the proton 180 pulse. However, the required phase mod-ulation can be achieved by adding a second proton 180 pulse at the end oft1, as shown in Fig. 1(a). This extra pulse effectively restricts the refocusingeffect of the first 180 pulse to the t1 period. From now on we will assumethat it is this slightly modified form of the HMBC experiment which is beingdiscussed.

The HMBC spectrum is invariant to the sign of the C–H coupling, so the fittingprocedure can give no information in this regard. Therefore, throughout thiswork we are only able to quote the magnitude of the coupling constant.

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2.1 The fitting procedure (Method I)

The way in which the fitting procedure works is best envisaged in the timedomain. If we imagine taking a cross-section parallel to F2 (the proton dimen-sion) through a cross peak in an HMBC spectrum, the time-domain signalcorresponding to the cross peak can be written as

SHMBC(t2) = AHMBC × Sproton(t2 + ∆) × sin (πJCHt2) (1)

where JCH is the long-range coupling responsible for the cross peak and AHMBC

is the overall amplitude; the factor sin (πJCH∆), which describes the depen-dence of the amplitude of the cross peak on the preparation delay, is includedin AHMBC. In Eq. 1 the term sin (πJCHt2) results in the observed multipletbeing anti-phase with respect to the active coupling JCH.

Sproton is the time-domain signal corresponding to the normal proton multiplet,with no C–H coupling present, such as would be acquired in a conventionalpulse–acquire experiment. This signal can be written as:

Sproton(t2) = exp (iΩH1t2) ×∏i

cos (πJH1Hit2) (2)

where ΩH1 is the offset of the proton involved in the cross peak (proton 1)and JH1Hi

is the coupling of proton 1 to proton i. The product of cosine termssimply expresses, in the time domain, the successive in-phase splittings thateach of these couplings causes. Note that, in contrast to the anti-phase C–Hcoupling, these in-phase H–H couplings are represented by cosine terms.

In the expression for SHMBC(t2), Eq. 1, the term Sproton(t2 + ∆) shows that atthe start of acquisition (i.e. t2 = 0) there has already been evolution of theproton offset and proton–proton couplings for time ∆.

The basis of our fitting method is the observation that the phase modulationdue to the evolution of the proton offset and proton-proton couplings duringdelay ∆ in the HMBC experiment is identical in form to that found for thesimple 90–∆–acquire(t2) sequence. The signal from this experiment can bewritten

S1D(t2) = Sproton(t2 + ∆). (3)

The only differences between S1D(t2) and the signal from the HMBC crosspeak (SHMBC(t2) of Eq. 1) are the amplitude and the presence of the sine termin the latter.

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The fitting procedure involves forming a trial multiplet by multiplying thetime-domain signal for the phase-modulated proton multiplet, S1D(t2) of Eq. 3,by a trial amplitude, Atrial, and a sine term to create an anti-phase splitting,sin (πJtrialt2):

Strial(t2) =Atrial × S1D(t2) × sin (πJtrialt2)

=Atrial × Sproton(t2 + ∆) × sin (πJtrialt2). (4)

We see that, provided Atrial = AHMBC and Jtrial = JCH, Strial(t2) becomesidentical to SHMBC(t2) of Eq. 1. This, then, is the basis of a fitting procedurein which Atrial and Jtrial are varied until Strial(t2) and SHMBC(t2) match asclosely as possible. This is achieved by minimizing the following χ2 function:

χ2 =

t2,max∫

0

|Strial(t2) − SHMBC(t2)|2 dt2. (5)

In Eq. 5 we have allowed for the fact that the signals are complex. For con-venience we refer to this original version of the fitting procedure as MethodI.

[Fig. 2 about here.]

It is convenient to perform the fitting routine in the time domain, but thewhole process can just as well be envisaged in the frequency domain; Fig. 2illustrates the fitting process in the two domains.

The form of the phase modulation due to proton offsets and proton–protoncouplings is only the same in the 90–∆–acquire(t2) and HMBC experimentsif the modified HMBC experiment of Fig. 1 (a) is used. It is not actually nec-essary to record the 90–∆–acquire(t2) experiment. Rather, we simply recorda conventional proton spectrum and then left shift the time-domain signal bya number of data points corresponding to the time ∆.

It is important to make sure that, for both the HMBC spectrum and thenormal proton spectrum, we deal with a single multiplet. This is achievedby excising the relevant multiplet from a one-dimensional spectrum or crosssection through a two-dimensional spectrum, by setting to zero all of the datapoints which do not define the multiplet. Having done this, the spectrum isinverse Fourier transformed to generate the time-domain signal, SHMBC(t2) orSproton(t2). Only then is the left shift applied to Sproton(t2) in order to createthe phase modulation.

As an absorption mode spectrum has the best resolution, it is usual to phasethe proton spectrum to absorption before the wanted multiplet is excised.

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However, although the real part of the (phased) spectrum is in the absorp-tion mode, the imaginary part will contain the much broader dispersion modemultiplet. To get around this problem we excise the absorption mode multi-plet from the real part of the spectrum and then generate the correspondingimaginary part using a Hilbert transform; further details can be found in ref-erence [5]. The resulting complex frequency-domain signal can then be inverseFourier transformed to give the required time-domain signal.

3 Introducing gradients into the HMBC experiment

For the fitting procedure to work as planned, it is clearly important that theHMBC spectra are of the highest quality. The use of pulsed field gradientsfor coherence selection is therefore to be recommended, as it is known thatspectra recorded in this way are generally superior to those recorded usingphase cycling.

There are many different ways in which gradients can be implemented intothe HMBC experiment, but for the fitting procedure described in the previoussection to be applicable, we have to be careful to ensure that the resultingmultiplets have exactly the same phase properties as those obtained from thesequence of Fig. 1(a).

One implementation which satisfies these requirements, and which we havefound to yield good spectra, is shown in Fig. 1(b). This sequence is the sameas that proposed by Cicero et al. [6] but with the addition of an extra 180

pulse on proton at the end of t1. As with sequence (a), this extra pulse isneeded so as to ensure that the proton evolution is identical to that of thesimple 90–∆–acquire(t2) experiment. In fact, because of the extra delays δneeded to accommodate the gradients, the proton offsets and couplings evolvefor a time (∆ + 2δ); this total delay must be used when constructing the trialmultiplet.

The 180 pulse on carbon allows selection of the required coherence order bythe two gradients G1 and G2; this pulse also refocuses any evolution of thecarbon offset which takes place during the two gradients. The gradient ratio,G1/G2, is set to −4.975/2.975 to record N -type data and −2.975/4.975 torecord P -type data. These two data sets are subsequently recombined in theusual way to give a frequency-discriminated, absorption mode spectrum [7].

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3.1 Phase correction

The presence during t1 of several pulses of finite duration makes it impossibleto acquire data for t1 = 0, leading to large frequency-dependent phase errorsin the F1 (carbon) dimension. While these can, in principle, be removed bymanual phasing in the usual way, such large linear phase corrections tend toresulted in distorted baselines. However, Zhu et al. [8] have shown that if thephase error across the spectrum is 180 or 360, subsequent correction doesnot lead to such baseline distortions. Thus, all that is required is that theinitial value of t1 be chosen to achieve such a phase error of 180 or 360.

Referring to Fig. 1 (b), the delays during which the carbon offset evolves can beidentified as: the first period t1/2, the duration of the first proton 180 pulse,the second period t1/2, and the duration of the second proton 180 pulse. Theevolution over the two delays δ is, to a good approximation, refocused by thecarbon 180 pulse.

Suppose that the initial value set for the delay t1/2 is t1,min/2. Then, for thefirst increment of the two-dimensional experiment, the time for which thecarbon offset evolves, t

(0)1 , is

t(0)1 = 2 × (t1,min/2) + 2 × t180, (6)

where t180 is the duration of the 180 pulses. Zhu et al. show that a linearphase correction of (n × 180) (where n is an integer) will be obtained if t

(0)1

satisfies

t(0)1 =

n

2δ1, (7)

where δ1 is the increment of t1, set by the required spectral width in F1. Inpractice, t1,min/2 is chosen so that Eq. 7 is satisfied. Finally, it should be notedthat as the time point t1 = 0 is not being sampled, it is not necessary to halvethe value of the first data point.

For example, in the experiments reported below the F1 spectral width of 181ppm (22624 Hz at a proton frequency of 500 MHz) gives a t1 increment, δ1,of 44.2 µs. The 180 pulse-length, t180, was 22.0 µs, so Eq. 7 can be satisfiedfor n = 2 by choosing t1,min/2 = 0.1 µs.

In F2 there is, in general, no phase correction which will result in absorptionmode multiplets; phase correction is, therefore, superfluous. However, it isdesirable to phase the proton spectrum from which the templates are to betaken so as to minimize the overlap between multiplets. If this is done, it is

7

essential to apply the same correction to the HMBC spectrum so that themultiplets are comparable. An alternative, which we have found convenient, isleave the HMBC spectra unaltered and to simply “undo” the phase correctionof the proton spectrum by applying to the template the opposite of the originalphase correction after the multiplet has been excised.

There is one further matter to take into account in respect of the phase inF2. This is the fact that, all other things being equal, there is a 90 phasedifference between the peaks in the HMBC and in the proton spectrum. Inthe simple proton experiment it is the in-phase magnetization which is theorigin of the observed signal. In contrast, in the HMBC experiment, the protonmagnetization which is anti-phase with respect to the 13C ultimately leads tothe observed signal. The evolution of in-phase magnetization (e.g. I1y) to anti-phase magnetization (e.g. 2I1xSz) involves a shift from the y- to the x -axis; thisis the origin of the 90 phase shift between the two spectra. In practice, thisphase shift is accounted for by multiplying the trial multiplet by exp (iπ/2) = iprior to the fitting i.e. Atrial should be purely imaginary.

[Fig. 3 about here.]

Figure 4 shows the result of the fitting procedure applied to the C8–H13 cross-peak multiplet from strychnine (structure and numbering shown in Fig. 3).With the correct choice of the two parameters, there is clearly an excellent fitbetween the trial multiplet and the multiplet from the HMBC. The contourplot of χ2, as defined in Eq. 5, as a function of the two parameters shows thatthe minimum is well defined so that it can easily be located by a simple searchprogram, such as the standard Levenberg–Marquardt algorithm [9].

[Fig. 4 about here.]

Typically, we construct this contour plot using quite a coarse grid of valuesfor the two parameters. This enables us to identify the approximate positionof the minimum, and then the values of the parameters at this point are usedas starting values for the Levenberg–Marquardt algorithm. Usually, this leadsto swift convergence.

The two modified versions of the fitting procedure described below involvefurther parameters, so it is not easy to visualize how χ2 depends on theirvalues. In such cases we use the best estimates we can for these additionalparameters and then plot how χ2 depends on Jtrial and Atrial. The position ofthe minimum identified in such a plot is used to obtain starting values for theLevenberg–Marquardt algorithm which is then allowed to alter the values ofall of the parameters. When using this approach, the minimum located by theLevenberg–Marquardt algorithm may deviate somewhat from that seen in theinitial χ2 contour plot.

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4 Using one-bond cross peaks as templates

It is often the case that cross peaks arising from one-bond C–H correlationsappear in HMBC spectra. These one-bond cross peaks have identical structureto the long-range cross peaks i.e. both are phase modulated by proton offsetsand couplings, and anti-phase with respect to the C–H coupling. However, asthe one-bond coupling is much larger than the width of the proton multiplet,the one-bond and long-range cross peaks look rather different. For the one-bond cross peak we see two clearly separated proton multiplets, separated bythe one-bond coupling and disposed symmetrically about the proton offset. Incontrast, the long-range cross peak appears as a single complex multiplet.

Sheng et al. [10] have described a version of the fitting procedure which usesthese one-bond cross peaks as the templates. Such an approach is certainlyattractive as all of the data needed to determine a value for the long-rangecoupling constant can be obtained from a single experiment. In addition, thereis likely to be less overlap of the multiplets than in the simple proton spectrum.However, there are two difficulties with using these one-bond cross peaks. Thefirst is that for a typical value of ∆ of around 50 to 100 ms, even the modestspread in the values of one-bond couplings results in a large variation in theintensity of the cross peaks. Indeed, it is quite common for a significant numberof the possible one-bond cross peaks to be absent. The second difficulty is thatthe one-bond cross peaks may overlap the long-range cross peaks, making itimpossible to use either for data fitting.

It would therefore be desirable to modify the HMBC experiment so that theone-bond cross peaks have reliable intensities, and so that the two types ofcross peaks can be separated. The latter aim can be achieved using the MBOBmethod developed by Schulte-Herbruggen et al. [11]. First, we will describethe basis of this method and then go on to describe how a similar approachcan be used to achieve more reliable intensities for the one-bond cross peaks.

4.1 Separating one-bond and long-range cross peaks

[Fig. 5 about here.]

The way in which the MBOB method works can be understood by referring toFig. 5. The thick and thin solid lines show how the amplitude of the one-bondcross peaks vary with the preparation delay, ∆, for two different values ofthe one-bond coupling constant; the rapid modulation is a result of the largecoupling constant. The grey line shows a similar curve for the much smallerlong-range coupling; in contrast, rather than an oscillation we just see a steadyrise. All the curves are plots of the function sin (πJ∆) — all that is different

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is the size of the coupling constant, J.

Consider recording two experiments with the preparation delays indicated bya and a′, separated by 1/(1JCH). Due to the properties of the sine wave, theamplitudes of the one-bond cross peaks in these two experiments will be equaland opposite. Thus, if the two experiments are added together, the one-bondcross peaks will cancel. In contrast, the amplitude of the long-range crosspeaks for these delays a and a′ are both positive and more or less the same;adding the two experiments together will reinforce these cross peaks. On theother hand, if the two experiments are subtracted, the one-bond cross peakswill be retained and the long-range cross peaks will more or less cancel. Withthis approach, separate one-bond and long-range spectra can be obtained fromthe same two data sets.

Figure 1 (c) shows how Schulte-Herbruggen et al. implement this idea by intro-ducing a mobile 13C 180 pulse at the start of the sequence. The preparationdelay ∆ is extended by a time ∆0 and the extra 180 pulse is placed during∆0 and at a time τf after the start of the sequence; τf can vary between zeroand ∆0.

For this simplest MBOB experiment (called a first-order filter) ∆0 is set to1/(2 1JCH) a delay which, for compatibility with the subsequent discussion, wewill call τ1. Two separate experiments are recorded for each value of t1: in thefirst τf = τ1 = 1/(2 1JCH); in the second τf = 0.

When τf = 0 the C–H coupling evolves for the whole time (∆ + τ1). Whenτf = τ1, the 180 pulse forms a spin echo which refocuses the evolution ofthe coupling at a time 2τ1 after the start of the sequence; thus the C–Hcoupling evolves for time (∆ − τ1). The effective time for which the couplingevolves in the two experiments thus differs by 2τ1 which, as τ1 = 1/(2 1JCH),is the required difference of 1/(1JCH). The advantage of varying the effectivepreparation delay in this way is that the proton offsets and the proton–protoncouplings evolve for time (∆ + τ1) in both cases.

Expressed mathematically, the amplitudes of the cross-peaks in the two ex-periments vary as a function of the preparation delay in the following ways:

Expt. 1 : sin (πJ(∆ − τ1)) Expt. 2 : sin (πJ(∆ + τ1)) (8)

where Expt. 1 and Expt. 2 correspond to τf = τ1 and τf = 0, respectively.The sum and difference of these two can be written, using some elementarytrigonometry, as:

Sum = Expt. 1 + Expt. 2 = 2 sin (πJ∆) cos (πJτ1)

Difference = Expt. 1 − Expt. 2 = −2 cos (πJ∆) sin (πJτ1).(9)

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For the long-range cross peaks, (πJτ1) is a very small angle so cos (πJτ1) ≈ 1and sin (πJτ1) ≈ 0. As a result, the long-range cross peaks appear only in thesum spectrum. For the one-bond cross peaks, (πJτ1) is approximately (π/2),so cos (πJτ1) ≈ 0 and sin (πJτ1) ≈ 1. As a result, the one-bond cross peaksappear only in the difference spectrum.

The separation is not perfect in part due to the range of values taken by theone-bond C–H coupling making it necessary to choose a compromise valueof τ1. Schulte-Herbruggen et al. show that the separation can be improvedby constructing higher-order MBOB filters. For example, a second-order filterinvolves recording four experiments in which the delay τf takes the successivevalues (0, τ1, τ2, τ1 + τ2), and where the delay ∆0 is (τ1 + τ2). The values of τ1

and τ2 are given by

τ1 =1

2[1JCH,min + 0.146(1JCH,max − 1JCH,min)](10)

τ2 =1

2[1JCH,max − 0.146(1JCH,max − 1JCH,min)], (11)

where 1JCH,min and 1JCH,max are the lower and upper bounds of the expectedrange of one-bond coupling constants. The sum of all four experiments containsjust the long-range cross peaks, whereas the one-bond correlations appear inthe combination of experiments (1 − 2 − 3 + 4).

4.2 Generating more reliable intensities for one-bond cross peaks

The problem with the intensities of the one-bond cross-peaks is illustratedclearly by Fig. 5. It would be quite possible to choose accidentally a prepara-tion delay which resulted in zero (or close to zero) intensity for a particularone-bond cross peak. Of course, a small change in the delay would result inthis cross-peak gaining significant intensity, but then another cross-peak aris-ing from a different value of the one-bond C–H coupling constant might havezero intensity. In general, there is no way of choosing a value of the preparationdelay such that all one-bond cross peaks will appear with significant intensity.

The broadband MBOB experiment [11] involves adding together, in absolutevalue mode, experiments recorded using different values of the preparationdelay, and so the one-bond cross peaks are present with reliable intensities.However, such an approach is not suitable for our purposes as phase-sensitivedata is needed by the fitting procedures.

Our solution to the problem of generating reliable intensities for one-bondcross peaks uses an approach similar to the MBOB filter. Two experiments

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are recorded with values of the preparation delay which differ by 1/(2 1JCH).Suppose the two delays correspond to b and b′, shown in Fig. 5. For a one-bondcoupling constant of 135 Hz, shown by the thick solid line, the amplitude ispositive when the delay is b and negative when the delay is b′. Thus, we wouldobtain greater overall amplitude by subtracting the two experiments. On theother hand, for a one-bond coupling constant of 125 Hz, shown by the thinsolid line, the amplitudes are positive at both b and b′, so the appropriateaction would be to add the two experiments together.

In general, as the value of the one-bond coupling is unknown, we cannotdetermine in advance whether to add or subtract the two spectra. Rather,both combinations are computed, and then the most intense cross peaks areselected from one or the other. To determine the separation of times b and b′

we use an average value of the one-bond coupling constant, as in the first-orderMBOB filter.

The experimental implementation of this approach is the same as for theMBOB experiment, Fig. 1 (c). However, for the present purpose, the extradelay ∆0 needs to be 1/(4 1JCH); to avoid confusion with the previous sectionwe will call this delay τ0. Two experiments are recorded, one with τf = τ0 andone with τf = 0. Thus, in the two experiments the C–H coupling evolves for(∆ − τ0) and (∆ + τ0), respectively.

Following the same analysis as before, the amplitudes of the one-bond crosspeaks in the sum and difference of the two experiments are

Sum =2 sin(π 1JCH∆

)cos

(π 1JCHτ0

)

Difference =−2 cos(π 1JCH∆

)sin

(π 1JCHτ0

). (12)

As τ0 = 1/(4 1JCH), cos (π 1JCHτ0) and sin (π 1JCHτ0) are both 1/√

2:

Sum =√

2 sin(π 1JCH∆

)

Difference =−√

2 cos(π 1JCH∆

). (13)

As was mentioned above, whether the sum or difference has the greater signaldepends on the precise values of the one-bond coupling and the preparationdelay, ∆. However, the key point is that this approach guarantees that allof the one-bond cross peaks will appear in either the sum or the differencespectrum with reasonable intensities.

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4.3 Combining the two approaches

The separation of the one-bond and long-range cross peaks by a first-orderMBOB filter can be achieved at the same time as generating more reliableintensities for the one-bond cross peaks by recording four separate experimentsin which the C–H coupling is allowed to evolve for the effective times, ∆eff ,given in the following Table 1.

Table 1Values of ∆eff and τf needed for a first-order MBOB filter,together with more reliable intensities for one-bond cross

peaks, in the modified HMBC sequence of Fig. 1 (d)

experiment ∆eff τf

1 ∆ + τ0 + τ1 0

2 ∆ − τ0 + τ1 τ0

3 ∆ + τ0 − τ1 τ1

4 ∆ − τ0 − τ1 τ0 + τ1

As before, τ0 = 1/(4 1JCH) and τ1 = 1/(2 1JCH). These four experiments canbe achieved experimentally using the pulse sequence shown in Fig. 1 (d) withthe value of the delay τf from the table and with ∆0 set to (τ0 + τ1).

The combination (1 + 2 + 3 + 4) can be shown to have the following variationof cross-peak intensity

4 sin (πJ∆) cos (πJτ0) cos (πJτ1). (14)

This is the combination in which, on account of the last term, the one-bondcross-peaks are suppressed. The long-range peaks are present at full intensityas, for small couplings, the last two terms are both very close to one.

There are two combinations in which the long-range cross peaks are sup-pressed. The combination (1 + 2− 3− 4) has the following variation of cross-peak intensity

4 cos (πJ∆) cos (πJτ0) sin (πJτ1), (15)

whereas the combination (1 − 2 − 3 + 4) has the following variation

4 sin (πJ∆) sin (πJτ0) sin (πJτ1). (16)

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These two combinations have little intensity from the long-range cross-peakson account of the sin (πJτ1) and sin (πJτ0) terms. As before, which experimenthas the greatest intensity for the one-bond cross peaks depends on the exactvalues of the one-bond coupling and the delay ∆.

[Fig. 6 about here.]

The separation of the long-range and one-bond cross peaks can be improvedby using a second-order MBOB filter. If this is combined with our methodfor giving more reliable intensities for the one-bond cross peaks, a total ofeight separate experiments have to be recorded with the values of τf given inTable 2; ∆0 is set to (τ0 + τ1 + τ2).

Table 2Values of ∆eff and τf needed for a second-order MBOB filter,

together with more reliable intensities for one-bond crosspeaks, in the modified HMBC sequence of Fig. 1 (d)

experiment τf experiment τf

1 0 5 τ0

2 τ1 6 τ0 + τ1

3 τ2 7 τ0 + τ2

4 τ1 + τ2 8 τ0 + τ1 + τ2

The sum of all eight experiments gives a spectrum in which only the long-range cross peaks are present. The other two combinations which contain onlythe one-bond cross peaks are : (1− 2− 3 + 4 + 5− 6− 7 + 8), the intensity inwhich depends on sin (πJ∆), and (1− 2− 3 + 4− 5 + 6 + 7− 8), the intensityin which depends on cos (πJ∆).

Figure 6 shows spectra of strychnine recorded using this approach. The spectrademonstrate both the clean separation of long-range and one-bond cross peakswhich can be obtained, and the different intensities of the cross peaks in thetwo combinations (b) and (c) in which only one-bond cross peaks appear.

5 Fitting using one-bond templates

As has been mentioned earlier, multiplets from the one-bond cross peak canbe used as templates in the fitting procedure so as to obtain estimates ofthe long-range couplings. There are two ways to proceed: the first, originallydescribed by Sheng et al. [10], is to use one half of the one-bond cross peak asa template; the second, described here for the first time, is to use the wholecross peak as a template.

14

The two halves of the one-bond cross peak are separated by 1JCH, which ismuch greater than the typical width of a proton multiplet. We might thereforeexpect that the two halves can be separated cleanly from one another. Thisis the case if the multiplet is in absorption mode, but in HMBC spectra themultiplets are in anything but absorption mode. When dispersion contribu-tions are present, significant overlap between the two sides of the one-bondcross peak can be seen, as shown by the fact that the baseline between themdoes not return to zero (see, for example, Fig. 10 (a)). In such circumstancesan entirely clean separation of the two halves is not possible and this is whyit may be preferable to use the entire one-bond cross peak.

5.1 Fitting using one half of the one-bond cross peak: Method II

Both halves of the one-bond cross peak already have the necessary phasemodulation due to the evolution of proton offsets and proton–proton cou-plings during the preparation delay. So, all that we need to do is excise oneside or the other, and then shift it by half of the one-bond coupling so thatthe excised multiplet is centred at the proton shift. Then the anti-phase cou-pling is introduced to create the trial multiplet which can be compared to theHMBC multiplet, just as before. Shifting the multiplet by half of the one-bondcoupling is performed in the time-domain.

[Fig. 7 about here.]

The overall fitting process can be described in the following way and is alsoillustrated in Fig. 7. The time-domain signal corresponding to a one-bondcross peak can be written

Sone−bond(t2) = Aone−bond × Sproton(t2 + ∆) × sin(π 1JCHt2

). (17)

The final term is the one which gives rise to the large anti-phase splitting bythe one-bond coupling, 1JCH.

In order to separate out the two sides of the multiplet we write the sine interms of exponentials:

sin(π 1JCHt2

)=

1

2i

[exp

(iπ 1JCHt2

)− exp

(−iπ 1JCHt2

)]. (18)

Selecting the right or left part of the multiplet involves selecting just one ofthese terms; arbitrarily we shall select the first one. Then the time domainsignal becomes

15

Shalf one−bond(t2) =1

2iAone−bond × Sproton(t2 + ∆) × exp

(iπ 1JCHt2

). (19)

The shift required to centre the multiplet at the proton offset is (−π 1JCH)rad s−1; this is achieved in the time domain by multiplying by exp (−iπ 1Jtrialt2).We have allowed for the fact that we may not know the value of the actualone-bond coupling, so will need to use a trial value, 1Jtrial.

The final step is to introduce the anti-phase long-range coupling by multiplyingby sin (πJtrialt2) and, as before, introducing an amplitude factor:

Strial(t2) =Atrial × Shalf one−bond(t2)

× sin (πJtrialt2) × exp(−iπ 1Jtrialt2

)

=1

2iAtrial Aone−bond × Sproton(t2 + ∆) × sin (πJtrialt2)

× exp(iπ 1JCHt2

)× exp

(−iπ 1Jtrialt2

). (20)

This function for the trial multiplet is compared with that for the long-rangecross peak (Eq. 1) in a least-squares fitting procedure. The parameters to beadjusted are the trial amplitude, the trial long-range coupling and the trialone-bond coupling. In practice, we have found that it is sufficient to estimatethe one-bond coupling from the spectrum and then just use this value for1Jtrial without further adjustment.

[Fig. 8 about here.]

Figure 8 shows this version of the fitting procedure in action for the C8–H13cross peak in strychnine. As before, with the correct choice of parameters, thetrial multiplet is an excellent fit for the long-range cross-peak multiplet.

5.2 Fitting using the entire one-bond cross peak: Method III

The second way of using the one-bond cross peak involves using the wholecross peak, rather than just half of it. In contrast to the previous case, boththe one-bond and long-range cross peaks are manipulated.

[Fig. 9 about here.]

The fitting procedure involves splitting the long-range multiplet by a trialanti-phase one-bond coupling, 1Jtrial, and splitting the whole one-bond crosspeak by a trial anti-phase long-range coupling, Jtrial. The two resulting multi-plets are then compared in a least-squares fitting procedure with the overall

16

amplitude and the two trial couplings being the adjustable parameters. Thewhole process is illustrated in Fig. 9.

The fitting process can be described mathematically in the following way.The long-range multiplet is described by the function SHMBC(t2) in Eq. 1;this function is multiplied by sin (π 1Jtrialt2) to generate the large anti-phasesplitting

S ′HMBC(t2) =SHMBC(t2) × sin

(π 1Jtrialt2

)

=AHMBC × Sproton(t2 + ∆)

× sin (πJCHt2) × sin(π 1Jtrialt2

). (21)

The one-bond multiplet is described by the function Sone−bond(t2) of Eq. 17.This function is multiplied by sin (πJtrialt2), to generate the small anti-phasesplitting; a trial amplitude is also included

S ′one−bond(t2) = Atrial × Sone−bond(t2) × sin (πJtrialt2)

= Atrial Aone−bond × Sproton(t2 + ∆)

× sin(π 1JCHt2

)× sin (πJtrialt2). (22)

We now see that provided the parameters Atrial, Jtrial and 1Jtrial have theappropriate values, S ′

HMBC(t2) and S ′one−bond(t2) are identical. The optimum

values of these parameters are found in a least-squares fitting procedure, asdescribed above. As before, we have found it sufficient to estimate 1Jtrial fromthe spectrum and then leave this parameter fixed.

[Fig. 10 about here.]

Figure 10 shows this version of the fitting procedure in action for the C8–H13cross peak in strychnine. As before, with the correct choice of parameters, thetwo manipulated multiplets are in close agreement.

5.3 Isotope shifts

When using one-bond cross peaks as templates in the fitting process, it isimportant to recognise that a one-bond and a long-range cross peak involvingthe same proton may not be centred at precisely the same offset in F2. Thisis because when the proton is directly attached to a 13C atom, as it is inthe molecules which lead to the one-bond cross peak, its chemical shift maybe slightly different to the case where the same proton is attached to a 12C

17

atom, as it is for molecules which lead to the long-range cross peak. This smalldifference is called an isotope shift.

The presence of this isotope shift means that the trial multiplet will never quitematch the HMBC multiplet no matter what values of the trial amplitude andcoupling are used. However, it is a simple matter to shift the offset of the trialmultiplet by multiplying it in the time domain by exp (iεt2), where ε is a trialshift which may be positive or negative. This shift simply becomes anotherparameter in the least-squares fitting procedure, and so its value is optimisedalong with the values of the other parameters. Typically, we have found thatincorporating this shift significantly improves the fit between the trial andHMBC multiplets. Shifts of up to 1 Hz are not uncommon.

It should be noted that in fitting method II, which uses half of the one-bondcross peak, introducing this parameter to allow for isotope shifts is identicalto allowing variations in the value of the trial one-bond coupling.

The contour maps of χ2 as a function of Atrial and Jtrial shown in Figs. 8 and10 are constructed using estimated values of the one-bond coupling and zeroisotope shift. The position of the minimum in these plots does not necessarilycorrespond, therefore, to the position of the minimum which will be foundwhen all of the parameters are allowed to vary. For example, in the caseof Fig. 10, the χ2 plot shows a minimum at Jtrial = 7.0 Hz, but when theisotope shift is allowed to vary, the Levenberg–Marquardt algorithm finds theminimum to be at 6.4 Hz.

6 A one-dimensional experiment

From a two-dimensional HMBC experiment we should be able to measure avalue for the coupling constant from the majority of the cross peaks. Havingsuch complete data from one experiment is an advantage, but there may beoccasions when only a small part of the data is actually required — for exam-ple, where a question of stereochemistry can be resolved by measuring just onelong-range coupling. In such situations, a selective one-dimensional version ofthe HMBC experiment may well be an attractive option as it can be recordedquickly and conveniently.

A large number of such selective experiments have been proposed over theyears [12]. For our present purposes, the main requirement is that the fittingprocedure developed for the two-dimensional HMBC can be applied to datafrom the one-dimensional experiment. A pulse sequence which satisfies thisrequirement is shown in Fig. 1 (e).

18

This sequence is essentially an HMBC experiment in which t1 has been re-placed by a selective echo on 13C; only long-range couplings to the carbon(or carbons) which experience the selective 180 pulse will give rise to observ-able signals. All other signals are suppressed effectively by the two gradientswhich flank the selective pulse. The proton offset and proton–proton couplingsevolve for the total time (∆ + 2δ + tsel), where the time periods are defined inFig. 1 (e).

Ideally, one would like to decouple the protons during the selective pulse to 13C.However, we have found it difficult to achieve this without at the same timedisrupting severely the evolution of the proton magnetization. In principle,it should be possible to minimize this effect by applying complete cycles ofa cyclic decoupling sequence such as WALTZ-16 [13]; we have had limitedsuccess with such an approach.

Since the selective pulse is applied to the proton-coupled 13C spectrum, thebandwidth of the pulse must be set to encompass a multiplet split by thelarge one-bond C–H coupling, unless it is a quaternary carbon which is beingexcited. It is possible to set the selective pulse to excite part of the multiplet(e.g. half of a doublet), although this will result in a loss of intensity.

The selective pulse is on resonance for the 13C of interest, so it is not necessaryto be concerned about the evolution of the offset. However, the pulse itselfmay give rise to a phase shift and this in turn will lead to a phase shift ofthe observed proton multiplet; clearly such a phase shift must be avoided asits presence would invalidate the fitting procedure. Our experience has beenthat with modern spectrometers, which are generally capable of attenuatingthe radio-frequency power level without introducing significant phase shifts,no problems associated with the phase of this selective pulse were found. Ifsuch phase errors are a problem, then a solution would be to use a double echo[14].

[Fig. 11 about here.]

Figure 11 shows three multiplets all arising from long-range correlations be-tween H8, H23b and H23a to C12 in strychnine. Spectrum (a) was obtainedusing the one-dimensional pulse sequence of Fig. 1 (f) whereas (b) was ob-tained using the modified HMBC sequence of Fig. 1 (e). The data show that,as is required, the one-dimensional experiment produces essentially identicalmultiplets to those found in the two-dimensional HMBC. The multiplets fromthe one-dimensional experiment can therefore be analysed in exactly the sameway as those from the two-dimensional experiment.

19

Table 3. Values of long-range C–H couplings (in Hz)in strychnine determined using different fitting procedures

C label H labela method I method III method II literature

15 13 3.5 3.4 6.6c

14 4.6 4.7 4.7c

8 6.3 6.4 6.3

12 0.6 0.8 1.2b

21 7.8 7.5 7.4b

14 15b 3.2 3.6 4.8c

13 3.3 3.5 3.8b, 4.6c

16 2.7 3.0

21 6.0 6.0 6.1b

7 17 3.2 † 3.1 3.2c

16/8 5.6 5.5

6 1.9 † 1.8

14 15a 1.8 2.8 2.8 3.2c

13 8.0 8.1 7.9c,7.7b

7 7.2 7.3 7.3c

16 4.5 4.3

12 0.9 2.3 2.1

21 0.9 1.4 1.9b

12 11b 7.0 6.9 7.1b

10 7.9 7.9 7.4b

14 20b 5.4 5.3 5.5c

18 3.5 3.1 3.4 3.6c

16 6.9 6.9

22 4.5 4.6

21 2.4 1.0 1.0

17 18b 5.6 5.1 4.9c

20 7.1 7.1 7.4c

13 11a 3.4d 3.3

12 2.5d 0.8 0.8 1.7b

10 5.8d 6.5 5.6 6.4b

17 18a 2.7 3.2 3.5 2.4c

7 4.6 5.2 5.2c

16 † 4.3

18 20a 9.3 9.6 9.5 9.5c

22 6.1 5.8

21 4.9 4.6

17 8 5.9 5.9 5.9c

7 2.5 2.4 2.6c

12 5.5 5.6 5.4b

6 3.7 3.7

5 3.2 3.1

12 23a 5.6 5.7

22 3.8 4.0

21 3.3 3.2

12 23b 8.6 8.2

22 4.0 3.9

21 8.5 8.3

8 12 5.8 5.9 5.8

23 2.4 6.8 2.4

10 † † 5.8

14 22 7.9 8.5 8.6 8.9c

20 4.9 5.6 5.1 12.5c, 5.7b

23 5.7 7.1 7.1 6.2b

2 4 7.5 7.4

6 5.5 5.5

5 0.9 0.8

adiastereotopic pairs labelled according to reference [15]. baverage value ofresults quoted in reference [16]. ctaken from reference [15]. dtemplate takenfrom TOCSY spectrum. †analysis failed to converge to a satisfactory result.

7 Results and discussion

Table 3 presents values for long-range couplings in strychnine which have beenmeasured using method I (taking the template from the proton spectrum) andmethod III (using the whole one-bond cross peak as the template); a smallernumber of couplings have also been measured using method II (using half theone-bond cross-peak as the template). Where they are available, the values ofthese couplings reported in the literature are also given.

Generally speaking, for couplings greater than about 2.5 Hz, there is goodagreement between the values derived from the three methods; we can thushave confidence in all of these methods. For smaller couplings, there tend to besignificant differences between the values determined by the different methods.Such small couplings give rise to weaker cross peaks and, in addition, the anti-phase splitting starts to become comparable with the linewidth. As a result,the minimum in the χ2 map becomes shallower so that the precise locationof the minimum itself is more easily perturbed by noise or experimental im-perfections; this is what accounts for the variation in the results given by thedifferent methods. For this data set, which is typical of what one might expect,we conclude that it would be unwise to interpret, in a quantitative way, anyvalues below approximately 2.5 Hz. This limit can be lowered by improvingthe signal-to-noise ratio (for example, by increasing ∆) and, if possible, theresolution.

The shape of the χ2 map is also influenced by the form of the HMBC multiplet.For example, multiplets which have complex structures, such as that shownin Fig. 4 (b), tend to give χ2 maps in which there is a single well-definedminimum. We can understand this by realizing that it is only when the valueof the trial coupling is correct that all of the details will match between theHMBC and trial multiplet. On the other hand, if the HMBC multiplet israther featureless, the minimum in the χ2 map is less well defined and so theresulting value of the coupling is less reliable.

[Fig. 12 about here.]

Occasionally, the χ2 plot does not show one well-defined minimum, such asthose seen in Figs. 4, 8 and 10. For example, fitting the C10–H12 cross peakusing method I gives the χ2 plot shown in Fig. 12. The global minimum is atJtrial ≈ 0.5 Hz, but there are subsidiary minima at 5.8 Hz and 10.5 Hz. Giventhe intensity of the cross-peak, it is inconceivable that the coupling can be assmall as 0.5 Hz, so despite this corresponding to the global minimum, we areforced to reject the value.

If the trial multiplets are inspected by eye it is found that for Jtrial = 0.5 Hzthere is quite close agreement with the HMBC multiplet, but that the fit does

21

not appear to be so good when Jtrial = 5.8 Hz. A further check is providedby comparing the C10–H12 cross peak with the C8–H12 cross peak. For thelatter, the fitting procedure straightforwardly gives a value of around 5.8 Hzfor the long-range coupling. However, this cross peak has little resemblance tothe C10–H12 cross peak; this casts doubt on the C10–H12 coupling being 5.8Hz as indicated by the subsidiary minimum in the χ2 plot. The lesson to bedrawn from this discussion is that when multiple minima are present, cautionis needed in interpreting the results of the fitting procedure.

In general, our conclusions about these fitting procedures are that: (1) reason-ably large values of the coupling constant determined by the fitting proceduresare reliable, but that small values need to be treated with caution; (2) it isuseful to fit a given cross peak using more than one method – the resultingspread of values gives some indication of reliability; (3) the χ2 plot should bechecked for the presence of multiple minima as these may be an indication ofpoor reliability.

Marquez et al. [16] have recently reviewed the large number of methods whichhave been proposed for measuring long-range C–H couplings. Most of thesemethods are based on HMQC- or HSQC-type correlation experiments whichhave been modified to manipulate the way in which either or both the proton-proton couplings and the long-range C–H coupling appear in the multiplets.The general aim is to achieve a multiplet from which a splitting can be iden-tified as the long-range C–H coupling. In this way, the need for any kind offitting procedure, such as that described here, is obviated. However, this isachieved at the expense of more complex pulse sequences and possible com-promises in the sensitivity of the experiment.

The procedure proposed in this paper relies on the use of a fitting procedure,and so the data analysis is inherently more complex than simply measuringa splitting from a multiplet. In addition, the fitting procedure requires extradata, but we have shown that by utilizing the one-bond cross peaks this extradata can be acquired at the same time as the HMBC spectrum. The pulsesequences are not complex and involve no compromises. Thus, the approachsuggested here represents a straightforward and efficient way of measuringvalues for long-range C-H couplings.

Acknowledgements

We are grateful to Keith Burton (Eli Lilly and Company) and Michael Thrip-pleton (University of Cambridge) for their interest in and helpful advice con-cerning this work. RAEE wishes to thank the EPSRC and Eli Lilly and Com-pany for a CASE studentship.

22

References

[1] A. Bax, M. Summers, 1H and 13C assignments from sensitivity-enhanceddetection of heteronuclear multiple-bond connectivity by 2D multiple quantumNMR, J. Am. Chem. Soc. 108 (1986) 2093–2094.

[2] J. J. Titman, D. Neuhaus, J. Keeler, Measurement of long-range heteronuclearcoupling constants, J. Magn. Reson. 85 (1989) 111–131.

[3] L. Braunschweiler, R. R. Ernst, Coherence transfer by isotropic mixing:application to proton correlation spectroscopy, J. Magn. Reson. 53 (1983) 521–528.

[4] M. J. Thrippleton, J. Keeler, Elimination of zero-quantum interference in two-dimensional NMR spectra, Angew. Chem., Int. Ed. Engl. 00 (2003) 00–00.

[5] R. N. Bracewell, The Fourier transform and its applications, McGraw-Hillinternational book company, London, 1978.

[6] D. O. Cicero, G. Barbato, R. Bazzo, Sensitivity enhancement of a two-dimensional experiment for the measurement of heteronuclear long-rangecoupling constants, by a new scheme of coherence slection by gradients., J.Magn. Reson. 148 (2001) 209–213.

[7] J. Ruiz-Cabello, G. W. Vuister, C. T. W. Moonen, P. van Gelderen, J. S. Cohen,P. C. M. van Zijl, Gradient-enhanced heteronuclear correlation spectroscopy -theory and experimental aspects, J. Magn. Reson. 100 (1992) 282–302.

[8] G. Zhu, D. A. Torcia, A. Bax, Discrete Fourier transformation of NMR signals.The relationship between sampling delay time and spectral baseline, J. Magn.Reson. Ser. A 105 (1993) 219–222.

[9] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numericalrecipes in C: the art of scientific computing, Cambridge University Press,Cambridge, 1996.

[10] S. Sheng, H. van Halbeek, Accurate and precise measurement of heteronuclearlong-range couplings by a gradient-enhanced two-dimensional multiple-bondcorrelation experiment, J. Magn. Reson. 130 (1998) 296–299.

[11] T. Schulte-Herbruggen, A. Meissner, A. Papanikos, M. Meldal, O. W. Sørensen,Optimising delays in the MBOB, broadband HMBC, and broadband XLOCNMR pulse sequences, J. Magn. Reson. 156 (2002) 282–294.

[12] T. Parella, High-quality 1D spectra by implementing pulsed-field gradients asthe coherence pathway selection procedure, Magn. Reson. Chem. 34 (1996)329–347.

[13] A. J. Shaka, J. Keeler, T. Frenkiel, R. Freeman, An improved sequence forbroadband decoupling: WALTZ-16, J. Magn. Reson. 52 (1983) 335–338.

23

[14] K. Stott, J. Stonehouse, T. L. Hwang, J. Keeler, A. J. Shaka, Excitationsculpting in high-resolution NMR spectroscopy – application to selective NOEexperiments, J. Am. Chem. Soc. 117 (1995) 4199–4200.

[15] V. Blechta, F. del Rıo-Portilla, R. Freeman, Long-range carbon–protoncouplings in strychnine, Magn. Reson. Chem. 32 (1994) 134–137.

[16] B. L. Marquez, W. H. Gerwick, R. T. Wiliamson, Survey of NMR experimentsfor the determination of nJ(C,H) heteronuclear coupling constants in smallmolecules, Magn. Reson. Chem. 39 (2001) 499–530.

24

1H

13C δ δ

G

φ1

φrx

∆

τf

1H

13C

∆

t1

t1

t2

t2

t2

t2

G2G1

G2G1

G2G1

tsel

1H

13C δ δ

G

∆

1H

13C

∆∆0

∆0

τf

(a)

(b)

(c)

(d)

t12

t12

1H

13C δ δ

G

∆(e)

Fig. 1. Pulse sequences for different variants of the basic HMBC experiment. Ra-diofrequency pulses applied to proton and carbon are shown on the lines marked1H and 13C, respectively; filled rectangles indicate 90 pulses, and open rectan-gles indicate 180 pulses. Pulse field gradients are shown in the line marked G.Sequence (a) is the conventional HMBC experiment, except that an extra proton180 pulse is included at the end of t1; see text for details. Sequence (b) is a gra-dient-selected HMBC experiment which has been designed specifically to producemultiplets with the same phase properties as those from sequence (a). The delays δare included simply to accommodate the gradient pulses G1 and G2. Sequence (c)shows how evolution of the C–H coupling can be affected by the placement of anextra 13C 180 pulse at the start of the sequence; see text for details. Sequence (d) isa new HMBC-type experiment which makes is possible to separate long-range fromone-bond correlations, and also results in more even intensities for the one-bondcross peaks. As described in the text, several data sets are recorded with differentvalues of τf and then recombined to give the required spectra. In sequence (d) thephases are as follows: φ1 = (x,−x) and φrx = (x,−x); all other pulses are phase x.Sequence (e) is a one-dimensional (selective) version of the HMBC experiment; theexperiment is designed to give multiplets with identical phase properties to thosefrom sequences (a), (b) and (d). The selective element is a 180 pulse, duration tsel,flanked by the two gradients.

25

frequency domain time domain

trial multiplet

left shift by ∆

multiply by Atrial sin(πJtrialt2)

2πJtrial

HMBC multiplet

compare compare

⊗

proton multiplet

Fig. 2. The basic fitting procedure (method I), described in detail in the text, can beenvisaged in either the frequency domain (shown on the left) or time domain (shownon the right). At the top is the conventional absorption-mode proton multiplet. Thefirst step is to introduce the phase modulation due to the preparation delay, ∆, in theHMBC sequence; it is convenient to do this by left shifting the time-domain signal bya number of data points corresponding to the delay ∆. Next, the anti-phase splittingis introduced: in the frequency domain this is achieved by convoluting (representedby ⊗) the proton multiplet with an “anti-phase stick multiplet” formed from twodelta-functions separated by 2πJtrial rad s−1; in the time domain the same effectis achieved by multiplication by Atrial sin (πJtrialt2). These two steps generate atrial multiplet which can then be compared with the multiplet from the HMBCspectrum; the two parameters Atrial and Jtrial are adjusted in a least-squares fittingprocedure in order to obtain the best fit between the trial and HMBC multiplets.In the diagram the trial multiplet has been constructed using the values of thesetwo parameters which give the best fit. Note that the time- and frequency-domainsignals are both complex, but that only the real parts are plotted here.

26

N

OO

N

12

3

4 5

6 7

89

10

11

12

1314

1516

1718

19

20

21

22

23

Fig. 3. The structure of strychnine, showing the numbering convention used.

27

HMBC multiplet

trial multiplet

proton multiplet

1.01.11.21.31.41.51.6 ppm

(a)

(b)

(d)

(c)

Jtrial / Hz

4 356789101112

-6

-8

-10

-4

-2

10

8

6

4

2

0

Atrial

Fig. 4. Experimental data showing application of the fitting procedure (method I)to the strychnine C8–H13 cross-peak multiplet. The H13 multiplet shown in (a) wasexcised from the conventional proton spectrum; the multiplet in (b) was excised froma cross section, taken parallel to F2, of the HMBC spectrum, through the C8–H13cross peak. Shown in (c) is the best-fit trial multiplet constructed as described inthe text; the agreement with (b) is excellent. Shown in (d) is a contour plot of χ2

(in arbitrary units), as defined in Eq. 5, as a function of the two parameters Atrial

and Jtrial. The minimum is clearly visible at Jtrial = 6.3 Hz.

28

a a'b b'

1/(1J )1/(2 1J )

10

ampl

itude

/ ar

b. u

nits

20 30 5040 60∆ / ms

Fig. 5. Plot showing how the amplitude of one-bond and long-range cross peaksvaries with the preparation delay, ∆. The thick and thin solid lines shows howthe amplitude of one-bond cross peaks vary for couplings of 135 Hz and 125 Hz,respectively. The grey line shows the slower variation which is characteristic of thevery much smaller long-range coupling (here 8 Hz). The significance of the timeslabelled a, a′, . . . is explained in the text.

29

2345 ppm

ppm

F2

F1

2345 ppm

30

40

50

60

70

80

2345 ppm

(a) (b) (c)

Fig. 6. Part of the HMBC spectra of strychnine showing the clean separation oflong-range and one-bond cross peaks, and the more reliable intensities of the latter,which can be obtained using the new method proposed here. Spectrum (a) is the sumof the eight separate experiments, recorded using the pulse sequence of Fig. 1 (d)and the values of τf given in Table 2; only long-range cross peaks are present. Spectra(b) and (c) are the two combinations (of the eight experiments) which contain justthe one-bond cross peaks; the different intensities of particular cross peaks in thetwo spectra are clearly visible. For each of the eight separate experiments, t1 wasincremented in 128 steps to a maximum value of 5.7 ms, and 8 transients wererecorded for each t1 value. The acquisition time was 0.8 s, and the recycle delaybetween experiments was 2.5 s. The delays ∆, τ0, τ1 and τ2 were 51.2, 1.750, 3.073and 3.928 ms, respectively; ∆0 was therefore 8.751 ms. The gradients G1 and G2

were of duration 1.95 ms, of Gaussian shape and truncated at the 1% level; theiramplitudes were set to −29.75% and 49.75% (of the full intensity of 40 G cm−1)for the P -type and −49.75% and 29.75% for the N -type spectrum. The delay δ was2 ms. Spectra were recorded at 500 MHz for proton, the total experiment time was13 hours and the sample concentration was approximately 120 mM in CDCl3.

30

take cross section

excise left-hand multiplet

trial multiplet

shift by π 1J

2π 1J

2πJtrial

⊗

Fig. 7. Illustration of a method, introduced by Sheng et al. [10], by which a one-bondbond cross peak can be used to generate a trial multiplet; we term this approachmethod II. First, a cross section containing the one-bond multiplet is taken fromthe two-dimensional spectrum; note that the phase modulation due to the evolutionof proton shifts and couplings is already present. Then, one half of the multipletis excised (here the left-hand side) and shifted to the right by half the one-bondcoupling; this centres the multiplet at the proton offset. Finally, the shifted multipletis convoluted with an anti-phase stick multiplet (as in Fig. 2) so as to introduce thetrial anti-phase coupling.

31

HMBC multiplet

trial multiplet

half of one-bond multiplet

1.01.11.21.31.41.51.6 ppm

(a)

(b)

(c)

(d)

Jtrial / Hz

4 3 2567891011

-6

-8

-10

-4

-2

10

8

6

4

2

0Atrial

Fig. 8. Experimental data showing method II being used to determine the value ofthe long-range C–H coupling from the C8–H13 cross-peak multiplet in strychnine.In contrast to Fig. 4, rather than using a multiplet from the proton spectrum, halfof the one-bond cross peak is used to construct the trial multiplet. Spectrum (a)shows the right-hand half of the one-bond cross peak (i.e. that between C13 andH13) which has been excised from a cross-section through the modified HMBCspectrum. Spectrum (b) shows the HMBC (long-range) multiplet which is to befitted, and (c) shows the best fit trial multiplet constructed from (a); the agreementbetween (b) and (c) is excellent. Also shown is a plot of χ2 as a function of the twoparameters Atrial and Jtrial; to compute this plot 1Jtrial is set to 125 Hz and the trialisotope shift, ε, is set to zero.

32

excise

2π 1J

excise

2π 1Jtrial

compare

2πJtrial

⊗ ⊗

Fig. 9. Illustration of fitting method III in which a one-bond cross peak is usedto generate a trial multiplet. On the left are shown the manipulations which areapplied to the whole one-bond cross peak. First the multiplet is excised from asuitable cross-section from the two-dimensional spectrum. Then, the trial anti-phaselong-range coupling is introduced, as before, by convolution. On the right are shownthe manipulations which are applied to the long-range cross peak. The multiplet isexcised and then the one-bond coupling is introduced. The resulting multiplets atthe bottom on the left and the right can now be compared and the best-fit valuesof Jtrial, 1Jtrial and Atrial found.

33

45 236789101112

(a)

(b)

(c)

(d)

1.01.11.21.31.41.51.6 ppm

one-bond multiplet

Jtrial / Hz

-6

-8

-10

-4

-2

10

8

6

4

2

0

Atrial

Fig. 10. Experimental data showing fitting method III being used on the C8–H13cross-peak multiplet from strychnine. In contrast to Fig. 8, rather than using oneside of the one-bond cross-peak multiplet, the whole multiplet is used. Spectrum(a) shows the complete one-bond multiplet and spectrum (b) is the long-range mul-tiplet into which the one-bond anti-phase coupling has been introduced. Spectrum(c) shows the effect of introducing the anti-phase splitting due to the long-rangecoupling into the multiplet shown in (a). At the optimum value of the long-rangecoupling (used here to construct (c)), there is excellent agreement between (b) and(c). Also shown is a plot of χ2 as a function of the two parameters Atrial and Jtrial;to compute this plot 1Jtrial is set to 125 Hz and the trial isotope shift, ε, is set tozero.

34

(a)

(b)

3.73.83.94.04.14.24.3 ppm

Fig. 11. Cross-peak multiplets arising from long-range correlations to C12 in strych-nine. Spectrum (a) was obtained using the one-dimensional pulse sequence ofFig. 1 (e) whereas (b) was obtained from a two-dimensional HMBC experiment.Note that, as required, the multiplets from the one-dimensional experiment are es-sentially identical to those from the two-dimensional HMBC. The parameters for thetwo-dimensional experiment are as in the caption to Fig. 6. For the one-dimensionalexperiment 1024 transients were recorded using an acquisition time of 0.8 s and arecycle delay between experiments of 1 s. The delays ∆ and δ were 55 and 2 ms,respectively. The gradients G1 and G2 were of duration 1.95 ms, of Gaussian shapeand truncated at the 1% level; their amplitudes were set to 29.76% and 49.76% (ofthe full intensity of 40 G cm−1). The selective 180 pulse was of duration 5 ms, ofGaussian shape and truncated at the 1% level. The total experiment time was 31minutes.

35

4 3 2 156789101112

-6

-8

-10

-4

-2

10

8

6

4

2

0

Jtrial / Hz

Atrial

1

2

3

3

44

4

5

55

Fig. 12. Contour plot of χ2 found when fitting the C10–H12 multiplet from strych-nine using method I. Contours are plotted at evenly spaced intervals and are labelled1, 2, . . . as the level rises. Three minima, indicated by •, are clearly visible; see textfor discussion.

36